Fractions in ratio problems
Ratios and fractions are two views of the same idea. Get fluent with switching between them and a whole class of "share in the ratio…" problems become easy.
Ratio basics — recap
A ratio compares two or more quantities. Read 3:5 as "3 parts to 5 parts" — the colon is just a separator. Ratios can usually be simplified by dividing every part by the same factor (their HCF).
12:18 simplifies (divide by 6) to 2:3.
Converting ratio → fraction
If a quantity is split in the ratio a:b, the total number of parts is a + b.
- The first share is a/(a+b) of the whole.
- The second share is b/(a+b) of the whole.
Worked example: divide £40 in the ratio 3:5.
- Total parts = 3 + 5 = 8 → each part = £40 ÷ 8 = £5.
- Shares: 3 × £5 = £15 and 5 × £5 = £25. (As fractions of the whole: 3/8 and 5/8.)
Three-way ratios
For a ratio a:b:c, total = a + b + c, and each share is the relevant numerator over that total.
Worked example: divide 720 sweets in the ratio 1:2:3.
- Total parts = 6. Each part = 120.
- Shares: 120, 240, 360. (Fractions: 1/6, 2/6 = 1/3, 3/6 = 1/2.)
Converting fraction → ratio
If A is 2/5 of the total, then B is 3/5 (the rest), and the ratio A:B = 2:3.
Worked example: girls make up 3/8 of a class. What is the ratio of girls to boys?
- Boys are 1 − 3/8 = 5/8.
- Girls : boys = 3/8 : 5/8 = 3:5.
Ratios that aren't already in lowest terms
When you're given e.g. "the ratio of cats to dogs is 12:8", simplify before treating it as a fraction.
12:8 → divide by 4 → 3:2. Cats are 3/5 of the cats+dogs total.
"Given one share" or "the difference" problems
These are the most common Higher-tier ratio questions. Strategy: find the value of one part first, then build up the rest.
Worked example: A and B share money in the ratio 3:7. B receives £36 more than A. How much does each receive?
- Difference in parts = 7 − 3 = 4 parts.
- 4 parts = £36 → 1 part = £9.
- A: 3 × £9 = £27. B: 7 × £9 = £63.
- Total = £90.
Worked example: A and B share sweets in the ratio 2:5. A receives 14 sweets. How many do they share in total?
- 2 parts = 14 → 1 part = 7.
- B has 5 parts = 35.
- Total = 14 + 35 = 49.
Ratios with fractional parts (e.g. converting to whole numbers)
If you're given a ratio with fractions, e.g. 1/2 : 1/3, multiply through by the LCM of the denominators.
1/2 : 1/3 × 6 → 3 : 2.
Combining two ratios — chaining
If A:B = 2:3 and B:C = 4:5, you need to make B match in both ratios.
- A:B = 2:3 = 8:12.
- B:C = 4:5 = 12:15.
- So A:B:C = 8:12:15.
This is a classic Higher-tier challenge.
⚠Common mistakes— Common mistakes (examiner traps)
- Treating one share as the total. "A gets 3/5" means 3 of every 5 parts, not "share so A has £3 out of £5".
- Forgetting to simplify before reading off the fraction.
- Confusing "more than" with "in total". The 4-parts difference vs 10-parts total trip lots of students up.
- Mixing up units — convert to the same units before forming a ratio.
- Direct addition of two ratios — to combine A:B and B:C, scale each so the shared term matches.
➜Try this— Quick check
A bag contains red, blue and green counters in the ratio 4:3:5. There are 60 counters in total. How many are blue?
- Total parts = 12. Each part = 60 ÷ 12 = 5.
- Blue = 3 × 5 = 15.
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